(i) Prove Taylor's Theorem for a function differentiable times, in the following form: for every there exists with such that
[You may assume Rolle's Theorem and the Mean Value Theorem; other results should be proved.]
(ii) The function is twice differentiable, and satisfies the differential equation with . Show that is infinitely differentiable. Write down its Taylor series at the origin, and prove that it converges to at every point. Hence or otherwise show that for any , the series
converges to .